\documentclass{article}

% American Mathematical Society
\usepackage{amsmath}

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\usepackage{geometry}

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\geometry{a4paper, scale=0.9}

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\pagestyle{empty}

\begin{document}
	% set the font size to large
	\large
	% set the line spacing to 4em
	\setlength{\baselineskip}{4em}
	
	% dx or d[#1]
	\renewcommand{\d}[1][x]{\ \text{d}#1}
	% for shorten minus delta
	\newcommand{\mdelta}{4 a c - b^2}
	% for shorten x^2 + \frac{b}{a} x + \frac{c}{a}
	\newcommand{\denA}{x^2 + \frac{b}{a} x + \frac{c}{a}}
	% for shorten (x + \frac{b}{2 a})^2 + \frac{c}{a} - \frac{b^2}{4 a^2}
	\newcommand{\denB}{(x + \frac{b}{2 a})^2 + \frac{c}{a} - \frac{b^2}{4 a^2}}
	
	\begin{align*}
		&\ \int \frac{mx + n}{a x^2 + b x + c} \d \ \ (a \neq 0)
		\\
		= &\ \int \frac{\frac{m}{a} x + \frac{n}{a}}{\denA} \d
		\\
		= &\ \int \frac{\frac{m}{a} x + \frac{n}{a}}{\denB} \d
		\\
		= &\ \frac{m}{a} \int \frac{x}{\denB} \d + \frac{n}{a} \int \frac{1}{\denB} \d
		\\
		= &\ \frac{m}{2 a} \int \frac{2 x + \frac{b}{a} - \frac{b}{a}}{\denB} \d + \frac{n}{a} \int \frac{1}{\denB} \d
		\\
		= &\ \frac{m}{2 a} \int \frac{2 x + \frac{b}{a}}{\denB} \d + \frac{2 n a - m b}{2 a^2} \int \frac{1}{\denB} \d
		\\
		= &\ \frac{m}{2 a} \int \frac{1}{\denA} \d[(\denA)] + \frac{2 n a - m b}{2 a^2} \int \frac{1}{(x + \frac{b}{2 a})^2 + (\sqrt{\frac{\mdelta}{4 a^2}})^2} \d[(x + \frac{b}{2 a})]
		\\
		= &\ \frac{m}{2 a} \ln{|\denA|} + \frac{2 n a - m b}{2 a^2} \frac{1}{\sqrt{\frac{\mdelta}{4 a^2}}} \arctan\frac{x + \frac{b}{2 a}}{\sqrt{\frac{\mdelta}{4 a^2}}} + C
		\\
		= &\ \frac{m}{2 a} \ln{|a x^2 + b x + c|} + \frac{2 n a - m b}{a\sqrt{\mdelta}} \arctan\frac{2 a x + b}{\sqrt{\mdelta}} + C
	\end{align*}
\end{document}
